Neumann (top) and Dirichlet (bottom) Boundary Conditions | ||||||||||||||
a1 + c1 | b1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u1 | f1 | ||
c2 | a2 | b2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u2 | f2 | ||
0 | c3 | a3 | b3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u3 | f3 | ||
0 | 0 | c4 | a4 | b4 | 0 | 0 | 0 | 0 | 0 | 0 | u4 | f4 | ||
0 | 0 | 0 | c5 | a5 | b5 | 0 | 0 | 0 | 0 | 0 | u5 | f5 | ||
0 | 0 | 0 | 0 | c6 | a6 | b6 | 0 | 0 | 0 | 0 | ´ | u6 | = | f6 |
0 | 0 | 0 | 0 | 0 | c7 | a7 | b7 | 0 | 0 | 0 | u7 | f7 | ||
0 | 0 | 0 | 0 | 0 | 0 | c8 | a8 | b8 | 0 | 0 | u8 | f8 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | c9 | a9 | b9 | 0 | u9 | f9 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | c10 | a10 | b10 | u10 | f10 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | c11 | a11 | u11 | f11 - b11u12 |
Step 1: Divide row 1 by its diagonal element.
1 | b1/(a1 + c1) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u1 | f1/(a1 + c1) | ||
c2 | a2 | b2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u2 | f2 | ||
0 | c3 | a3 | b3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u3 | f3 | ||
0 | 0 | c4 | a4 | b4 | 0 | 0 | 0 | 0 | 0 | 0 | u4 | f4 | ||
0 | 0 | 0 | c5 | a5 | b5 | 0 | 0 | 0 | 0 | 0 | u5 | f5 | ||
0 | 0 | 0 | 0 | c6 | a6 | b6 | 0 | 0 | 0 | 0 | ´ | u6 | = | f6 |
Step 2: Multiply row 1 by c2 and subtract the result from row 2.
1 | b1/(a1 + c1) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u1 | f1/(a1 + c1) | ||
0 | a2- c2b1/(a1 + c1) | b2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u2 | f2- c2f1/(a1 + c1) | ||
0 | c3 | a3 | b3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u3 | f3 | ||
0 | 0 | c4 | a4 | b4 | 0 | 0 | 0 | 0 | 0 | 0 | u4 | f4 | ||
0 | 0 | 0 | c5 | a5 | b5 | 0 | 0 | 0 | 0 | 0 | u5 | f5 | ||
0 | 0 | 0 | 0 | c6 | a6 | b6 | 0 | 0 | 0 | 0 | ´ | u6 | = | f6 |
Step 3: Divide row 2 by its diagonal element.
1 | b1/(a1 + c1) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u1 | f1/(a1 + c1) | ||
0 | 1 | b2(a1 + c1)/[a2(a1 + c1) - c2b1] | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u2 | [ f2(a1 + c1) - c2f1 ] / [a2(a1 + c1) - c2b1] | ||
0 | c3 | a3 | b3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u3 | f3 | ||
0 | 0 | c4 | a4 | b4 | 0 | 0 | 0 | 0 | 0 | 0 | u4 | f4 | ||
0 | 0 | 0 | c5 | a5 | b5 | 0 | 0 | 0 | 0 | 0 | u5 | f5 | ||
0 | 0 | 0 | 0 | c6 | a6 | b6 | 0 | 0 | 0 | 0 | ´ | u6 | = | f6 |
Step 4: Multiply row 2 by c3 and subtract the result from row 3.
Step 5: Etc., until you end up with a matix of the form ...
1 | b1' | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u1 | f1' | ||
0 | 1 | b2' | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u2 | f2' | ||
0 | 0 | 1 | b3' | 0 | 0 | 0 | 0 | 0 | 0 | 0 | u3 | f3' | ||
0 | 0 | 0 | 1 | b4' | 0 | 0 | 0 | 0 | 0 | 0 | u4 | f4' | ||
0 | 0 | 0 | 0 | 1 | b5' | 0 | 0 | 0 | 0 | 0 | u5 | f5' | ||
0 | 0 | 0 | 0 | 0 | 1 | b6' | 0 | 0 | 0 | 0 | ´ | u6 | = | f6' |
0 | 0 | 0 | 0 | 0 | 0 | 1 | b7' | 0 | 0 | 0 | u7 | f7' | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | b8' | 0 | 0 | u8 | f8' | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | b9' | 0 | u9 | f9' | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | b10' | u10 | f10' | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | u11 | f11' |